E19 -- De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt

(On transcendental progressions, that is, those whose general terms cannot be given algebraically)


One of Euler’s earlier papers, he starts with Wallis’ "hypergeometric series" 1! + 2! + 3! + 4! + ... (he didn’t have the !-factorial notation yet) to find the value of the function for a general value of x. He defines [x] to be ∫01 (ln(1/t))x dt, which by substituting t=e-z, equals ∫0zxe-zdz, or what we know as the Gamma function. He finds [1/2] to be √(p/2) and derives the recursion [x+1] = (x+1)[x]. He finds [p/q] to be a product of beta functions, and derives a differential quotient.

According to the records, it was presented to the St. Petersburg Academy on November 28, 1729.
Euler gave the essential content of this treatise to his friend Goldbach in a letter on January 8, 1730.

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